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Mirrors > Home > MPE Home > Th. List > prnz | Structured version Visualization version GIF version |
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.) |
Ref | Expression |
---|---|
prnz.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
prnz | ⊢ {𝐴, 𝐵} ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prnz.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | prid1 4698 | . 2 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | 2 | ne0ii 4303 | 1 ⊢ {𝐴, 𝐵} ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 {cpr 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4568 df-pr 4570 |
This theorem is referenced by: opnz 5365 propssopi 5398 fiint 8795 wilthlem2 25646 upgrbi 26878 wlkvtxiedg 27406 shincli 29139 chincli 29237 spr0nelg 43658 sprvalpwn0 43665 |
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